Step 1. Given:
-The square piece of paper that Kitan cut has a side length of 6 inches
-The sector of a circle that Mai cut has a radius of 6 inches and an arc length of 4pi.
Required: Find whose paper is larger.
We start by making a diagram of the circle sector and the square as shown in the image:
Step 2. To find which one is larger, we will need to find the area of the two figures.
First, we calculate the area of the square as follows:
[tex]A_{square}=(length)^2[/tex]Since the length is 6 in, the area is:
[tex]\begin{gathered} A_{square}=(6in)^2 \\ \downarrow \\ \boxed{A_{square}=36in^2} \end{gathered}[/tex]Step 3. To find the area of the circle sector, first, we have to find which part of a total circle the sector represents.
Let's find the circumference of the whole circle:
[tex]\begin{gathered} C=2\pi r \\ C=2\pi(6in) \\ C=12\pi in \end{gathered}[/tex]If we had a complete circle, the total arc or circumference would be 12pi:
But note that the sector is only 4pi, this means that the sector represents 1/3 of a complete circle.
Step 4. Once we know that the sector is one-third of a complete circle, we find its area by dividing the total area of the circle by 3:
[tex]A_{circle}=\frac{\pi r^2}{3}[/tex]Substituting the known values and using 3.14 as the value of pi:
[tex]A_{c\imaginaryI rcle}=\frac{(3.14)(6in)^2}{3}[/tex]Solving the operations:
[tex]\begin{gathered} A_{c\imaginaryI rcle}=\frac{(3.14)(36\imaginaryI n^2)}{3} \\ \downarrow \\ \boxed{A_{c\mathrm{i}rcle}=37.68in^2} \end{gathered}[/tex]Step 5. Comparing the two areas:
[tex]\begin{gathered} \boxed{A_{square}=36\imaginaryI n^{2}} \\ \boxed{A_{c\mathrm{\imaginaryI}rcle}=37.68\imaginaryI n^{2}} \end{gathered}[/tex]We can see that the area of the circle sector is larger, therefore Mai's paper is larger.
Answer: Mai's paper is larger
